Okay I'm telling everyone, this is why we need to discover mathematical equations in forming friendships/relationships, so that we can turn that "loneliness frown" upside down.

Historians have said what set Galileo/Newton apart from the others of their time was they found mathematical equations for their ideas. It helped them be creative, not in a sense of humor/entertainment way but rather scientific "come up with new ideas that are useful" way. Likewise if we could do the same with looking at data scatter plots for forming relationships/friendships, maybe it could allow us to be "creative" in a come up with new and useful ways of interacting with that special someone. Isaac Newton was creative, so why can't we?

You're right, correlation doesn't prove causation, but rather association. One possibility is to then control for other variables and see if the correlation still remains. Although it doesn't prove nor even probably cause-effect, it can give more/less confidence in a conclusion.

It's the same concept when the health field says smoking is bad for you. It's not ethical to give an experimental group so many smokes a day and a control group a placebo and see who is more likely to die earlier, however they have extensively tried controlling for other variables and found that the correlation regarding mortality rates among smokers is still there even after all that. Not as good as a true controlled experiment and can't prove, but does give more/less confidence.

In the case of this study, I'd guess one thing they'd need to control for is the possibility of lonely people hanging out with others who are already lonely vs. not already lonely. This could be done by comparing the lonely people who hang out with the non lonely people to see if the correlation is still when their friends are lonely later on also. I haven't looked at the actual study stated there and it could get extremely complex, so I have no idea what effort they put into internal validity, but you get my point.

I'm pretty sure it's yawning. It's something I noticed when I was on the lecture tour. First one person in the audience yawns and then they all start yawning. Some even go to sleep.

I have experienced the same effect, jimmy. The yawn reflex seemed somehow to be tied to steam-table training for boiler operators. They were probably just cooling their brains and signaling readiness. I was flattered.